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Existence of positive solutions of some semilinear elliptic equations with singular coefficients

Published online by Cambridge University Press:  12 July 2007

Nirmalendu Chaudhuri
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore 560012, India
Mythily Ramaswamy
Affiliation:
School of Mathematics, TIFR Centre, IISc, Bangalore 560012, India

Abstract

In this paper we consider the semilinear elliptic problem in a bounded domain Ω ⊆ Rn, where μ ≥ 0, 0 ≤ α ≤ 2, 2α* := 2(n − α)/(n − 2), f : Ω → R+ is measurable, f > 0 a.e, having a lower-order singularity than |x|-2 at the origin, and g : R → R is either linear or superlinear. For 1 < p < n, we characterize a class of singular functions Ip for which the embedding is compact. When p = 2, α = 2, fI2 and 0 ≤ μ < (½(n − 2))2, we prove that the linear problem has -discrete spectrum. By improving the Hardy inequality we show that for f belonging to a certain subclass of I2, the first eigenvalue goes to a positive number as μ approaches (½(n − 2))2. Furthermore, when g is superlinear, we show that for the same subclass of I2, the functional corresponding to the differential equation satisfies the Palais-Smale condition if α = 2 and a Brezis-Nirenberg type of phenomenon occurs for the case 0 ≤ α < 2.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2001

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